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StatsToDo: Sample Size for Comparing Two Normally Distributed Means
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Explanations and References
This page presents 4 programs related to sample size requirements when comparing 2 means, and tables of these sample sizes.
Sample Size Tables
The programs and tables on this page assumes that the measurements are continuous and normally distributed. The following 4 programs are available on this page
ReferencesArmitage P (1980) Statistics in Medical Research. Blackwell Scientific Publication, Oxford. ISBN 0 632 05430 1 p.120Machin D, Campbell M, Fayers, P, Pinol A (1997) Sample Size Tables for Clinical Studies. Second Ed. Blackwell Science IBSN 0-86542-870-0 p. 24-25 Johanson GA and Brooks GP (2010) Initial Scale Development: Sample Size for Pilot Studies. Educational and Psychological Measurement Vol.70,Iss.3;p.394-400
These tables provides sample sizes (per group) needed to compare the means of 2
groups, assuming that the data is normally distributed, that the variance
in the groups are similar, and that the sample sizes for the groups are the same
Javascript Programs
- Power = 1-β - α = Probability of Type I Error - es = Effect size = difference between means / population of within group Standard Deviation - column header 1T and 2T represents 1 and 2 tail - sample size is that for each of the 2 groups Effect Size 0.05 to 1
This subpanel presents 4 R programs related to sample size for 2 unpaired means
Program 1: Estimating sample size
# Pgm 1: Sample Size
# data entry
dat = ("
Alpha Power Diff SD
0.05 0.8 0.5 1.0
0.01 0.8 0.5 1.0
0.05 0.9 0.5 1.0
0.01 0.9 0.5 1.0
")
df <- read.table(textConnection(dat),header=TRUE) # conversion to data frame
# vectors to store results
SSiz1Tail <- vector()
SSiz2Tail <- vector()
# Calculations
delta <- abs(df$Diff / df$SD) # effect size
zb <- abs(qnorm(1 - df$Power)) # z for beta
# 1 tail
za <- abs(qnorm(df$Alpha)) # 1 tail z for alpha
f <- (za + zb) / delta
SSiz1Tail <- append(SSiz1Tail,ceiling(2.0 * f**2 + za**2 / 4.0)) # 1 tail sample size
# 2 tail
za <- abs(qnorm(df$Alpha / 2)) # 2 tail z for alpha
#za
f <- (za + zb) / delta
SSiz2Tail <- append(SSiz2Tail,ceiling(2.0 * f**2 + za**2 / 4.0)) # 2 tail sample size
# append results to data frame
df$SSiz1Tail <- SSiz1Tail
df$SSiz2Tail <- SSiz2Tail
df # show data frame with input data and rsults
The results are as follows. Sample size is for each group
> df # show data frame with input data and rsults Alpha Power Diff SD SSiz1Tail SSiz2Tail 1 0.05 0.8 0.5 1 51 64 2 0.01 0.8 0.5 1 82 96 3 0.05 0.9 0.5 1 70 86 4 0.01 0.9 0.5 1 106 121 Program 2: Power
# Pgm2: Power
# data entry
dat = ("
Alpha Diff N1 SD1 N2 SD2
0.05 0.5 64 1.0 64 1.0
0.01 0.5 96 1.0 96 1.0
0.01 0.5 64 1.0 96 1.0
0.05 0.5 86 1.0 86 1.0
0.01 0.5 121 1.0 121 1.0
0.01 0.5 86 1.0 121 1.0
")
df <- read.table(textConnection(dat),header=TRUE) # conversion to data frame
# vector to store results
Power1Tail <- vector()
Power2Tail <- vector()
# Calculations
se <- sqrt(((df$N1 - 1) * df$SD1**2 + (df$N2 - 1) * df$SD2**2) / (df$N1 + df$N2 - 2))
n <- (df$N1 + df$N2) / 2
delta <- abs(df$Diff / se)
# 1 tail
za <- abs(qnorm(df$Alpha)) # 1 tail z for alpha
zb <- delta * sqrt(n / 2.0 - za / 8.0) - za
Power1Tail <- append(Power1Tail,pnorm(zb)) # power 1 tail
# 2 tail
za <- abs(qnorm(df$Alpha / 2)) # 2 tail z for alpha
zb <- delta * sqrt(n / 2.0 - za / 8.0) - za
Power2Tail <- append(Power2Tail,pnorm(zb)) # power 2 tail
# append ewsults to data frame
df$Power1Tail <- Power1Tail
df$Power2Tail <- Power2Tail
df # show data frame with input data and results
The results are as follows
> df # show data frame with input data and results Alpha Diff N1 SD1 N2 SD2 Power1Tail Power2Tail 1 0.05 0.5 64 1 64 1 0.8798970 0.8044474 2 0.01 0.5 96 1 96 1 0.8701805 0.8096574 3 0.01 0.5 64 1 96 1 0.7951479 0.7169130 4 0.05 0.5 86 1 86 1 0.9480270 0.9048008 5 0.01 0.5 121 1 121 1 0.9398340 0.9036948 6 0.01 0.5 86 1 121 1 0.8962385 0.8437134 Program 3: Confidence Interval
# Pgm3: Confidence Interval
# subroutine to calculate confidence interval
# subroutine to calculate confidence interval
CalCI <- function(alpha, n1, sd1, n2, sd2)
{
se = sqrt(((n1 - 1) * sd1**2 + (n2 - 1) * sd2**2) /
(n1 + n2 - 2) * (1 / n1 + 1 / n2))
degfm = n1 + n2 - 2
t1 = qt(1 - alpha, degfm)
ci1 = t1*se
t2 = qt(1 - alpha / 2, degfm)
ci2 = t2 * se
return (c(se, ci1, ci2))
}
The main program for confidence interval, using the subroutine CalCI
# data entry
dat = ("
Pc N1 SD1 N2 SD2
95 100 18.5 100 16.8
99 100 18.5 100 16.8
95 50 8.5 30 5.2
99 50 8.5 30 5.2
95 50 400 60 380
")
df <- read.table(textConnection(dat),header=TRUE) # conversion to data frame
# df
# vectors for results
SE <- vector() # Standard Error
CI1 <- vector() # Confidence interval 1 tail
CI2 <- vector() # Confidence interval 2 tail
# calculations
# 1 tail
for(i in 1 : nrow(df))
{
alpha <- (1 - df$Pc[i] / 100) # alpha 1 tail
ar <- CalCI(alpha,df$N1[i], df$SD1[i], df$N2[i], df$SD2[i])
#ar <- CalCI(1 - df$Pc[i] / 100,df$N1[i], df$SD1[i], df$N2[i], df$SD2[i])
SE <- append(SE, ar[1]) # Standard Error
CI1 <- append(CI1, ar[2]) # Confidence interval 1 tail
CI2 <- append(CI2, ar[3]) # Confidence interval 2 tail
}
df$SE <- SE
df$CI1 <- CI1
df$CI2 <- CI2
df
The results are as follows
> df # Data frame withinput data and results Pc N1 SD1 N2 SD2 SE CI1 CI2 1 95 100 18.5 100 16.8 2.498980 4.129778 4.928032 2 99 100 18.5 100 16.8 2.498980 5.860928 6.499565 3 95 50 8.5 30 5.2 1.719553 2.862410 3.423366 4 99 50 8.5 30 5.2 1.719553 4.084128 4.540204 5 95 50 400.0 60 380.0 74.526406 123.645653 147.724266 Program 4: Pilot studyPlease Note: This program shares the same subroutine as Program 3 to calculate SE and CI, but the subroutine is not duplicated here
# Program 4. Pilot study
# Parameters
pc = 95 # % confidence
sd = 1.0 # within group or population SD
intv = 5 # interval
maxN = 100 # maximum sample size
# vectors for results
SSiz <- vector() # sample size
CI1 <- vector() # confidence interval 1 tail
Diff1 <- vector() # difference in CI from previous row 1 tail
DecCase1 <- vector() # decrease in CI per case increase 1 tail
PDCase1 <- vector() # % decrease in CI per case increase 1 tailCI1 <- vector() # confidence interval 1 tail
CI2 <- vector() # confidence interval 2 tail
Diff2 <- vector() # difference in CI from previous row 2 tail
DecCase2 <- vector() # decrease in CI per case increase 2 tail
PDCase2 <- vector() # % decrease in CI per case increase 2 tail
# Calculations
alpha = 1 - pc / 100
# first row
n = intv
SSiz <- append(SSiz,n)
ar <- CalCI(alpha, n, sd, n, sd)
ci1 = ar[2] * 2
CI1 <- append(CI1,sprintf(ci1, fmt="%#.4f")) # confidence interval 1 tail
Diff1 <- append(Diff1,0) # difference in CI from previous row 1 tail
DecCase1 <- append(DecCase1,0) # decrease in CI per case increase 1 tail
PDCase1 <- append(PDCase1,0) # % decrease in CI per case increase 1 tailCI1 <- vector() # confidence interval 1 tail
ci2 = ar[3] * 2
CI2 <- append(CI2,sprintf(ci2, fmt="%#.4f")) # confidence interval 1 tail
Diff2 <- append(Diff2,0) # difference in CI from previous row 1 tail
DecCase2 <- append(DecCase2,0) # decrease in CI per case increase 1 tail
PDCase2 <- append(PDCase2,0) # % decrease in CI per case increase 1 tailCI1 <- vector() # confidence interval 1 tail
# subsequent rows
while(n < maxN)
{
n = n + intv
SSiz <- append(SSiz,n)
ar <- CalCI(alpha, n, sd, n, sd)
oldci1 = ci1
ci1 = ar[2] * 2
CI1 <- append(CI1,sprintf(ci1, fmt="%#.4f")) # confidence interval 1 tail
diff1 = oldci1 - ci1
Diff1 <- append(Diff1,sprintf(diff1, fmt="%#.4f")) # difference in CI from previous row 1 tail
decCase1 = diff1 / intv
DecCase1 <- append(DecCase1,sprintf(decCase1, fmt="%#.4f")) # decrease in CI per case increase 1 tail
pDCase1 = sprintf(decCase1 / oldci1 * 100, fmt="%#.1f")
PDCase1 <- append(PDCase1,pDCase1) # % decrease in CI per case increase 1 tail
oldci2 = ci2
ci2 = ar[3] * 2
CI2 <- append(CI2,sprintf(ci2, fmt="%#.4f")) # confidence interval 2 tail
diff2 = oldci2 - ci2
Diff2 <- append(Diff2,sprintf(diff2, fmt="%#.4f")) # difference in CI from previous row 2 tail
decCase2 = diff2 / intv
DecCase2 <- append(DecCase2,sprintf(decCase2, fmt="%#.4f")) # decrease in CI per case increase 2 tail
pDCase2 = sprintf(decCase2 / oldci2 * 100, fmt="%#.1f")
PDCase2 <- append(PDCase2,pDCase2) # % decrease in CI per case increase 2 tail
}
df <- data.frame(SSiz,CI1,Diff1,DecCase1,PDCase1,CI2,Diff2,DecCase2,PDCase2)
df # display results in data frame
The results are as follows. Please note:
> df # display results in data frame SSiz CI1 Diff1 DecCase1 PDCase1 CI2 Diff2 DecCase2 PDCase2 1 5 2.3522 0 0 0 2.9169 0 0 0 2 10 1.5510 0.8012 0.1602 6.8 1.8791 1.0378 0.2076 7.1 3 15 1.2423 0.3087 0.0617 4.0 1.4959 0.3832 0.0766 4.1 4 20 1.0663 0.1760 0.0352 2.8 1.2803 0.2156 0.0431 2.9 5 25 0.9488 0.1175 0.0235 2.2 1.1374 0.1430 0.0286 2.2 6 30 0.8632 0.0856 0.0171 1.8 1.0337 0.1037 0.0207 1.8 7 35 0.7973 0.0659 0.0132 1.5 0.9540 0.0797 0.0159 1.5 8 40 0.7444 0.0528 0.0106 1.3 0.8903 0.0637 0.0127 1.3 9 45 0.7009 0.0435 0.0087 1.2 0.8379 0.0524 0.0105 1.2 10 50 0.6642 0.0367 0.0073 1.0 0.7938 0.0441 0.0088 1.1 11 55 0.6328 0.0315 0.0063 0.9 0.7560 0.0378 0.0076 1.0 12 60 0.6054 0.0274 0.0055 0.9 0.7231 0.0329 0.0066 0.9 13 65 0.5813 0.0241 0.0048 0.8 0.6942 0.0289 0.0058 0.8 14 70 0.5598 0.0214 0.0043 0.7 0.6685 0.0257 0.0051 0.7 15 75 0.5406 0.0192 0.0038 0.7 0.6454 0.0231 0.0046 0.7 16 80 0.5232 0.0174 0.0035 0.6 0.6246 0.0208 0.0042 0.6 17 85 0.5074 0.0158 0.0032 0.6 0.6057 0.0189 0.0038 0.6 18 90 0.4930 0.0144 0.0029 0.6 0.5883 0.0173 0.0035 0.6 19 95 0.4797 0.0133 0.0027 0.5 0.5724 0.0159 0.0032 0.5 20 100 0.4674 0.0123 0.0025 0.5 0.5578 0.0147 0.0029 0.5 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||